I think its Yuji from Shakugan no Shana. In the first season, he couldnt do much, but now he can cast a Fuzetsu and gained some other powers.

I agree.

There would be similar characters like probably Ichigo or the ladies in Magic Knight Rayearth, all who didn't know anything even about the existence of the other world at first, but at least in the case of Yuji, we can see the effort and the snail pace progress throughout the show. The others just sort of gain some powers, like shinigami powers for Ichigo, and suddenly learned how to fight.

In terms of skill, Naruto would probably be the worst at this. 200+ episodes later and still the only things he can do are Kage Bunshin and Rasengan (albeit more powerful). But not in terms of maturity though.